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Properties

Label 2-792-24.11-c1-0-39
Degree $2$
Conductor $792$
Sign $-0.475 - 0.879i$
Analytic cond. $6.32415$
Root an. cond. $2.51478$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.885 − 1.10i)2-s + (−0.430 − 1.95i)4-s − 3.67·5-s − 4.28i·7-s + (−2.53 − 1.25i)8-s + (−3.25 + 4.05i)10-s − i·11-s + 7.11i·13-s + (−4.72 − 3.79i)14-s + (−3.63 + 1.68i)16-s + 4.85i·17-s + 0.571·19-s + (1.58 + 7.17i)20-s + (−1.10 − 0.885i)22-s − 3.61·23-s + ⋯
L(s)  = 1  + (0.626 − 0.779i)2-s + (−0.215 − 0.976i)4-s − 1.64·5-s − 1.62i·7-s + (−0.895 − 0.444i)8-s + (−1.02 + 1.28i)10-s − 0.301i·11-s + 1.97i·13-s + (−1.26 − 1.01i)14-s + (−0.907 + 0.420i)16-s + 1.17i·17-s + 0.131·19-s + (0.353 + 1.60i)20-s + (−0.235 − 0.188i)22-s − 0.753·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.475 - 0.879i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.475 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.199862 + 0.335018i\)
\(L(\frac12)\) \(\approx\) \(0.199862 + 0.335018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.885 + 1.10i)T \)
3 \( 1 \)
11 \( 1 + iT \)
good5 \( 1 + 3.67T + 5T^{2} \)
7 \( 1 + 4.28iT - 7T^{2} \)
13 \( 1 - 7.11iT - 13T^{2} \)
17 \( 1 - 4.85iT - 17T^{2} \)
19 \( 1 - 0.571T + 19T^{2} \)
23 \( 1 + 3.61T + 23T^{2} \)
29 \( 1 + 3.99T + 29T^{2} \)
31 \( 1 + 7.49iT - 31T^{2} \)
37 \( 1 + 5.62iT - 37T^{2} \)
41 \( 1 + 4.14iT - 41T^{2} \)
43 \( 1 + 7.29T + 43T^{2} \)
47 \( 1 + 5.48T + 47T^{2} \)
53 \( 1 + 6.86T + 53T^{2} \)
59 \( 1 - 0.106iT - 59T^{2} \)
61 \( 1 - 4.52iT - 61T^{2} \)
67 \( 1 - 4.89T + 67T^{2} \)
71 \( 1 + 0.651T + 71T^{2} \)
73 \( 1 + 6.32T + 73T^{2} \)
79 \( 1 + 13.2iT - 79T^{2} \)
83 \( 1 + 7.98iT - 83T^{2} \)
89 \( 1 - 5.40iT - 89T^{2} \)
97 \( 1 + 1.44T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.996097028831753366405860670356, −8.962372961678032014846881396914, −7.895820496269387796971901224541, −7.10296610292484044983173698529, −6.22934684685605864339462030301, −4.59943661405064457455450246253, −3.97747072758986344176236606329, −3.63605781403530837490951476696, −1.71391918378225071756830800401, −0.15692596049690615449013949234, 2.85205139677255222444583203605, 3.43431161146910875084109413811, 4.86074009984457667445216653715, 5.34534172947988452433870942343, 6.48932824634056948037243207959, 7.55626288895217950521190280401, 8.130968092015955294320170907066, 8.696658534694992795048288025114, 9.845268748419023476120963027638, 11.29479742299331846518099840767

Graph of the $Z$-function along the critical line